Optical fiber based modal filtering sensors are suitable for overcoming observer spillover which is a phenomenon that a filtering sensor's output contains. Theoretically, any structure has an infinite number of vibration modes, but in vibration control, only a finite number of modes will be incorporated in the control algorithm. An observer is used in the control algorithm to obtain full states (parameters of the controlled modes) required by feedback when only partial states can be obtained from sensor outputs. This observer spillover contains the uncontrolled mode information which influences the observer and makes the controller unstable. Filtering sensors are used in conjunction with control algorithms for minimizing undesirable mechanical vibrations in flexible structures. When designing these vibration control algorithms, it has been discovered that a spatially distributed type sensor can detect information of all vibration modes of a structure that is experiencing full spectrum vibration and retain controlled modes and filter any uncontrolled modes present in the structure.
The theoretical basis of vibration modal filtering sensors taught by the prior art and used by the invention herein comes from propagation properties of light in weakly-guided optical fibers and the theory of fiber-optic interferometric sensors. As an example of such a sensor, a dual-mode fiber-optic sensor can be expressed by the phase difference (.DELTA..phi..sub.0) between the two propagating bees within an optical fiber as follows: EQU .DELTA..phi.=.DELTA..phi..sub.0 +.intg.[.DELTA..beta.(x).multidot..epsilon.(x,t)].delta.x for x between L.sup.- and L.sup.+ ( 1)
where .DELTA..phi..sub.0 is a constant for a given sensor, L.sup.- and L.sup.+ are the starting and ending positions of the sensor, .DELTA..beta.(x) is the difference of the propagation constants of the beams and is called the "weighting function", x is the coordinate along the fiber sensor and .epsilon.(x,t) is the strain applied to the fiber-optic sensor. The term .epsilon.(x,t) contains the information of various vibrating modes, and can usually be expressed as .epsilon.(x,t)=.SIGMA.Y.sub.i" (x).eta..sub.i (t) for i equal to zero to infinity, where Y.sub.i" (x) is related to the mode shape and .eta..sub.i (t) is known as the modal coordinate. If .DELTA..beta.(x) is designed such that .DELTA..beta.(x) and Y.sub.i (x) meet orthogonality conditions, only the i.sup.th mode information is retained in the sensor output, thus yielding a vibration modal filter. From the above analysis,this propagation constant, .DELTA..beta.(x) has to be varied along the length of the fiber sensor and "programmed" therein.
The governing equations for light propagating in a weakly guided optical fiber are as follows: EQU [uJ.sub.l.+-.l (u)]/[J.sub.l (u)]=[.gamma.K.sub.l.+-.l (.gamma.)]/ [K.sub.l (.gamma.)] where l=0,1, (2)
and EQU u.sup.2 +.gamma..sup.2 =V.sup.2 ( 3)
where
u.sup.2 =(n.sup.2.sub.o k.sup.2.sub.o -.beta..sup.2) a.sup.2 ; PA0 .gamma..sup.2 =(.beta..sup.2 -n.sup.2.sub.c k.sup.2.sub.o) a.sup.2 PA0 k.sub.o =2.pi./.lambda..sub.o and .lambda..sub.o is the light's wavelength in free space; PA0 a is the core radius and .beta. is the propagation constant; PA0 n.sub.c and n.sub.o are the refractive indices of the cladding and core respectively; PA0 V=k.sub.o [.sqroot.(n.sup.2.sub.o -n.sup.2.sub.c)] is known as the normalized frequency.
If V can be varied along the sensing fiber length, then u, .gamma. and .beta. from Eqs. (2) and (3) are functions of the coordinate along the sensing fiber length. Further, if .beta. can be a varied along the sensing fiber length, the expected .DELTA..beta. (x) in equ. (1) can be obtained by designing the normalized frequency V number along the sensing fiber length.
U.S. Pat. No. 5,224,182 entitled "Spatially-Weighted Two-Mode Optical Fiber Sensors" by Murphy et al. teaches of varying the fiber core diameter by a tapering method as a way of obtaining a modal filtering sensor. This teaching uses a fiber drawing tower to fabricate optical fiber sensors with known taper profiles by controlling the furnace temperature, the preform-feed speed and the fiber-pull speed. This method can also taper the fiber by using a coupler station. The basis of its' operation is the difference of the propagation constants of the two modes in a two-mode fiber and can use a weighting function for vibration filtering if it varies along the fiber length. Limitations of this type method of producing a modal filter include i) the fabrication thereof is difficult due to the small change of the fiber core diameter; ii) the complexity of the expected weighting functions; and iii) the degradation to fiber strength due to diameter change.
U.S. Pat. No. 5,208,877 by Murphy et al. entitled "Fiber Optic Grating-Based Weighted, Two-Mode Fiber Sensors" teaches of a two-mode optical fiber sensor with a permanent photo-induced index of refraction core change that can be used for modal distributed vibrational filtering applications using a Hill-like grating method. In particular, this teaching obtains a fiber-optic vibration mode filter for a cantilever beam by writing a Hill-like grating in a germanium-doped dual-mode optical fiber. The gratings occurred only at those locations where the two supported optical modes interfere with each other. These locations periodically position down the fiber length and are dependent on the strain in the fiber when light at the same wavelength used for writing the gratings at 488 nm is coupled into the fiber. The mode filter reflects all the light when the strain in the fiber matches the strain during the writing condition. The power transmitted by the fiber increases non-linearly as the strain gets further from the writing condition. To retain the first mode of the filtering and filter other mode information of a structure, for example using a cantilever beam in the sensor output, this teaching uses a Hill-like grating at the same time when the sensing fiber was prestrained in the shape of the first mode, and the linear region of the strain-power curve is used for performing sensor measurements. Limitations of this method of producing such a filtering sensor include: i) harmonic distortion caused by nonlinear behavior of the strain-power response; ii) theoretical models used for producing the same are very complex and not yet available; iii) the filtering sensor must be prestrained to the mode shape that is to be retained; iv) the grating writing process is based on interferometry which dictates stringent control over environmental disturbances during making of the filter; v) the wavelength of 488 nm used to write the gratings is far from the optimal wavelength of 244 nm; and vi) the filter must be interrogated using the same wavelength used to produce the gratings wherein this constraint is significant since argon-ion lasers used in the production of this filter are large and expensive, making their use in interrogating these Hill-like grating based modal filters impractical for field applications.
To overcome limitations of the above, the invention herein directly changes the refractive index profile of the optical fiber core for a single-mode interferometric based filtering device.